places of holomorphic function

If c is a complexconstant and f a holomorphic function in a domain D of ℂ, then f has in every compact (closed (http://planetmath.org/TopologyOfTheComplexPlane) and bounded (http://planetmath.org/Bounded)) subdomain of D at most a finite set of http://planetmath.org/node/9084c-places, i.e. the points z where f⁢(z)=c, except when f⁢(z)≡c in the whole D.

Proof. Let A be a subdomain of D. Suppose that there is an infinite amount of c-places of f in A. By http://planetmath.org/node/2125Bolzano–Weierstrass theorem, these c-places have an accumulation pointz0, which belongs to the closed setA. Define the constant functiong such that

g⁢(z)=c

for all z in D. Then g is holomorphic in the domain D and g⁢(z)=c in an infinite subset of D with the accumulation point z0. Thus in the c-places of f we have